1.1. Review by: Richard Bellman.
Quarterly of Applied Mathematics 19 (2) (1961), 110.

This is an excellent outline for an introductory text in the theory of ordinary differential equations. The only serious weakness as far as the outline is concerned is the lack of matrix techniques in dealing with N-th order linear equations and linear systems.

The author attempts to present in 147 pages, everything from basic concepts of point sets to linear equations to existence theory; from power series solutions to approximate solutions to Sturm-Liouville theory. It can't be done.

1.2. Review by: Pasquale Porcelli.
Mathematics Magazine 34 (1) (1960), 43.

This is an excellent text designed for a one semester course in ordinary differential equations and is written in a frank, clear, and lively manner. The author states at the outset that he is writing for students who are familiar with the elements of advanced calculus and, except for the second chapter (first order equations), makes effective use of the calculus throughout the book. In short the book is written in the refreshing spirit of modern mathematics and, unlike the usual run of textbooks, adopts the attitude that our students can catch up to Newton (and possibly go beyond him). There are nine chapters in the book. The first discusses basic concepts. The third and fourth deal with the second and $N$th order linear equations and make effective use of the Wronskian. Chapter five starts with a crystallisation of the metric space concepts (actually, the author sets up a flavour for this in the previous chapters) and presents two proofs to the fundamental existence theorem: the first is based on the classical Picard method and the second proof is obtained as a by-product of contraction mappings. The remaining chapters cover linear systems, special functions, approximate solutions, and a survey of well-selected topics. There are a large number of formal exercises throughout the book together with problems of a serious nature.

1.3. Review by: Peter L Balise.
American Scientist 48 (4) (1960), 362A.

Courses in differential equations have various objectives, such as reviewing techniques of solution, showing applications in science and engineering, or illustrating general mathematical concepts. This little book distinctly emphasises the latter, briefly presenting some topics that are usually not mentioned in texts on differential equations.

The first few pages define elementary concepts of set theory, and the rest of the work is similarly based on mathematical rigour. While this reduces the readability for the student who is not a mathematics major, it should be noted that mathematical logic is being increasingly applied in such engineering fields as system analysis.

The usual first-order equations and linear equations of higher order are considered in the first half of the book, followed by a relatively detailed chapter on existence theory. The presentation of systems is limited to two simultaneous linear equations, and the generalisation is effectively indicated by exercises for the student. There is brief discussion of linear equations with non-constant coefficients, and of special topics such as orthogonality and eigenfunctions. Non linear equations are only briefly mentioned, apparently because a general mathematical basis for their analysis is not available.

The omission of tools such as the phase plane illustrates the fundamental approach of this book. It achieves a good compact presentation of the mathematical basis of ordinary differential equations.

1.4. Review by: Arthur E Danese.
Amer. Math. Monthly 68 (2) (1961), 192.

If you have grown weary teaching the traditional "cook-book" course in differential equations, this is the book to consider. The usual content of a differential equations course - minus applications - is presented in a rigorous setting, requiring the student to exhibit understanding of point set theory and elementary real variable theory (in the text and in the variety of exercises). Two proofs of the fundamental existence theorem are presented - one employing the classical Picard iterative technique and the other an elegant approach through the use of functional analysis. In general, the book has a systematic and well-written approach to its subject. In particular, the Frobenius method for solution in series is admirably treated.

I have some reservations, however. I gather the impression from certain topics scantily outlined (special functions, Laplace transform, Sturm-Liouville theory) that the author is pressed for space. Yet he presents linear differential equations of order two in one chapter and then generalises them practically word for word to those of order n in the next chapter. Also, it is disturbing to find the author recalling the properties of determinants, but neglecting to state theorems involving interchange of limit operations. I might add that the la test editions (Agnew, Churchill, Zygmund) should be cited in the bibliography.

Nevertheless, there is a great deal-to praise in this book and it should prove a welcome relief for those who are interested in more than routine solution of differential equations without resort to treatises.

1.5. Review by: Mario O González.
Mathematical Reviews MR0123759 (23 #A1081).

This book offers a fresh approach to the traditional first course on ordinary differential equations.

In the first four chapters of the book the author does not depart much from the usual treatment, except that special types of equations and special methods of solution are reduced to a minimum. These chapters deal with basic concepts, first-order equations and linear differential equations of second and higher orders. The exposition is neat and rigorous. No applications to physics, chemistry, engineering, etc., are given.

In Chapter 5 the existence theory is developed. The classical Picard iteration procedure is discussed in detail and also the modern or functional analysis approach based on the theory of metric spaces and the Banach fixed-point principle.

In Chapter 6 linear systems are discussed briefly. In Chapters 7, 8 and 9 brief but adequate introductions are presented of the following topics: linear differential equations of the second order with variable coefficients leading to the definitions of some special functions, approximate solutions, orthogonality and Sturm-Liouville problems. This text may be used to advantage for a first course on differential equations for students of mathematics and physics.

1.6. Review by: Florence N David.
Biometrika 48 (1/2) (1961), 237-238.

With the growth of the 'dynamics' of statistical method commonly but not necessarily correctly lumped together as 'stochastic processes' the interest of the student of mathematical statistics in the theory of differential equations has perforce been sharpened. This book is written for those who know a reasonable amount of advanced calculus but not all of it will be useful to those who seek to acquire mathematical techniques rather than mathematical learning.

The topics covered are first-order equations, linear differential equations of the second and nth order, existence theory, linear systems, special functions, approximate solutions, eigenfunctions and Fourier series. The delineation is clear and the book is eminently readable. Exercises are given at the end of each chapter and the answers supplied. It should serve as a useful introduction to the subject for mathematicians. The 'user' student may find it too difficult.

2.1. From the publisher.

Designed for use in a 1-semester course by seniors and beginning graduate students, this rigorous presentation explores practical methods of solving differential equations, plus the unifying theory underlying the mathematical superstructure. Topics include basic concepts, Fourier series, 2nd-order partial differential equations, wave equation, potential equation, heat equation, and more. Includes exercises.

2.2. Review by: Helge Skovgaard.
Nordisk Matematisk Tidskrift 10 (4) (1962), 204-205.

The book is written as a beginner's book for mathematics, physics and engineering students and is intended to be used in a six-month course. The presentation, given in a clear and inciting style, and at the same time in a modern and exact form, makes the book easy to read. Given the choice of matter within the rather limited space, the author has wanted to steer directly towards some of physics' most well-known differential equations (wave equation, potential equation and thermal conductivity equation) and to illustrate how to solve initial and boundary value problems in this area both by exact and numerical methods, the latter particularly for the use of electronic calculators.
...

However, the overall impression of the book is absolutely favourable, and there is no doubt that those who are interested in this subject and would like to learn something about it in an easily accessible way will read the book with benefit and interest.

2.3. Review by: Edward T Copson.
The Mathematical Gazette 47 (359) (1963), 83-84.

This book is one of McGraw-Hill's International Series in Pure and Applied Mathematics, a series which includes books of distinction, such as Ahlfors's Complex Analysis. I cannot understand how Dr Greenspan's book got into such exalted company: it has nothing whatever to commend it. The plain fact is that the author does not know for whom he is writing. The pure mathematician would do better by going to the third volume of Goursat; the applied mathematician will not find here the way to solve the problems he meets every day.

Despite the use of very high-brow terminology, the only equations discussed are Laplace's equation, the wave equation and the equation of heat, in each case with only two independent variables. The publishers claim that "Modern high-speed computer methods are developed"; "developed" is an over-statement - all that appears is an introduction to the finite-difference approximation method.
...

And the book concludes with a pep talk; mathematicians are urged to support and encourage research in the theory of non-linear equations.

There is an interesting bibliography, which unfortunately contains references to multigraphed lecture notes not easily obtainable

2.4. Review by: Bernard A Fleishman.
SIAM Review 4 (3) (1962), 260-261.

In writing this book, the author seems to have had in mind two primary purposes: first of all, to base application and technique on rigorously developed theory; secondly, within the realm of technique, to give numerical methods the careful attention which is due them. He has succeeded very well in achieving these aims, and in the process has written a sophisticated, up-to-date text for a first course in partial differential equations. The reader is presumed to have a (solid) background in advanced calculus. ...
...

The presentation is formal and concise. Proofs are given in detail, but in the opinion of this reviewer the book would have benefited greatly from more ample non-rigorous discussion (e.g., of motivation, meaning of new concepts, significance of results). Thus, the instructor employing it in a course will have an essential role to play in motivating, interpreting and elaborating the text material.

These faults, however, are minor compared to the book's very substantial positive features. It represents a valuable addition to the resources of modern mathematics education.

2.5. Review by: Richard T Shield.
Quarterly of Applied Mathematics 22 (2) (1964), 162.

The Preface of this book asks the reader to note that "Chapters 1 and 2 merely develop those aspects of ordinary differential equations, complex variables, and Fourier series which are essential for the study of partial differential equations, which begins properly in Chapter 3." Since Chapters 1 and 2 occupy only sixty-five pages (four of these being devoted to ordinary differential equations), the reviewer cannot help but question the reasonableness of this request. The remaining one-hundred and thirty pages contain a poorly written and uninspired introduction to the theory of second order partial differential equations.

2.6. Review by: Nicholas D Kazarinoff.
Mathematical Reviews MR0130453 (24 #A314).

This book is intended for use in a one-semester course for either undergraduate or graduate students, for engineers or students of mathematical sciences. There are eight chapters, of which the first two on "basic concepts" and Fourier series are introductory. The next five deal only with equations of second order in exactly two variables, and the last is a cursory summary of four and one-half pages on three distinct subjects. Applications of the theory to problems of physics and engineering are neglected.

The exposition is generally clear, concise and rigorous. The discussion of the Dirichlet problem for Laplace's equation is done most thoroughly of all, and is well done. The chapter on the wave equation is good. Other topics included are the classification of equations of second order (and their canonical forms), characteristics, the heat equation, and numerical methods.

The reviewer is keenly disappointed by the author's decisions to limit so severely the class of equations he considers and to omit all discussion of generalised derivatives and generalized solutions.

2.7. Review by: Watson Fulks.
Amer. Math. Monthly 69 (5) (1962), 449.

This book is concerned with second-order equations in two variables. Chapter 1, Basic Concepts, concerns itself largely with the pertinent parts of advanced calculus. The amassing of such a phalanx of definitions as presented here is perhaps justified since most of them should be familiar to properly prepared students. Chapter 2 is an adequate discussion of Fourier series. In Chapter 3 the Cauchy problem is defined, characteristics discussed briefly, and linear equations with constant coefficients reduced to canonical forms. Chapter 4 discusses the wave equation. D'Alembert's solution of the Cauchy problem is obtained for both the homogeneous and the nonhomogeneous equations. A uniqueness proof for the initial-boundary value problem is given, and the Fourier series solution is given in the case of zero initial derivative and zero boundary values.

A discussion of Laplace's equation constitutes Chapter 5. A weak maximum principle is established, the Dirichlet problem is defined, a uniqueness proof given, and the solution by Perron's method is carried out for regions bounded by contours. Chapter 6 contains a short discussion of the heat equation. Again a weak maximum principle is presented, and is used to prove uniqueness for the initial -boundary value problem for a rectangle. A Fourier series solution of this problem is given in the case of zero boundary data. Chapter 7 contains a discussion of numerical methods, and Chapter 8 mentions the application of the Laplace transform and states the Cauchy-Kowalewski theorem.

The book is well written with copious examples and problems. I am somewhat concerned about the absence of any discussion of the Neumann problem, even in so short an introduction. (It is mentioned in an exercise.) Also, since the Perron method is essentially an interior method, the author's standard restriction to regions bounded by contours may be misleading. Such a condition could well be postponed until the assumption of the boundary values is discussed.

3.1. Review by: M A Feldstein.
Amer. Math. Monthly 74 (3) (1967), 344.

As the title implies this monograph introduces the reader to constructive solutions for elliptic problems. The opening chapter constitutes a dozen pages of prerequisite analytic concepts.

Through carefully prepared stages the author swiftly moves from an analysis of the standard two dimensional Laplace equation (Chapter 2) to multi-dimensional linear elliptic problems (Chapter 6). Along the way the reader is initiated to the mode of attack: a motivational development of appropriate difference equations followed by a convergence proof. (A ten page appendix analyses some iterative methods for actually solving both linear and nonlinear systems.)

The final chapter (7) is titled Nonlinear Problems. By its very nature the material here quickly reaches research frontiers. Various open problems are presented. Chapter 7 in particular concludes with a carefully referenced detailed discussion of four "Research Problems."

Algorithms are clearly labelled (called "Methods"); theorems and lemmas on the whole are concisely stated, a rather unusual feat in numerical analysis; numerical examples are presented in a form which one can assimilate readily without becoming overwhelmed by a surplus of digits. Motivation is provided by associating the various differential equations with physical phenomena. The author continually teaches that problems need be well posed. There is an abundance of exercises and a very extensive bibliography.

It is unfortunate that the author chose to omit any analysis of rate of convergence or of error propagation - at either the discretisation or iteration phases. Otherwise, this excellent text is admirably suited for use with either advanced undergraduates or beginning graduates. Though it will likely appeal primarily to mathematics majors, it should certainly be recommended for suitable serious students of the sciences. The price (about 44 cents per page) seems almost double what would be reasonable.

3.2. Review by: Eugene Isaacson.
Mathematical Reviews MR0179956 (31 #4193).

The book is intended as a supplementary text for a college course on numerical methods. The chapter headings will indicate the scope of this brief work: (1) Analytical preliminaries; (2) The Dirichlet problem for the two-dimensional Laplace equation; (3) The mixed problem for the two-dimensional Laplace equation; (4) Dirichlet and mixed problems for more general two-dimensional linear elliptic equations; (5) Three-dimensional problems with axial symmetry; (6) Linear elliptic problems in three and more dimensions; (7) Nonlinear problems.

The book has an excellent bibliography. The emphasis of the book is on proving the convergence of the solution of the difference equations to the solution of the differential equation as the mesh width is decreased. The analysis of several iterative methods for solving the difference equations is relegated to a brief appendix and is intentionally cursory.

4.1. Review by: Olof Widlund.
Mathematics of Computation 24 (110) (1970), 487-488.

This book is a survey of numerical methods for the solution of differential equations, based on the author's lectures at summer conferences at the University of Michigan. According to the preface, "Scientists and technologists should be able to determine easily from the text what the latest methods are and whether those methods apply to their problems". There are 486 references that "will enable teachers to adapt the material for classroom presentation".

The reviewer would hesitate very much to use this book as a textbook or to suggest it as reading for applied scientists asking for advice on numerical methods. The material is quite specialised, and only a few methods are discussed. No proofs are given. In the discussion of elliptic problems, the problem of convergence and accuracy is not even mentioned, and the reader is left without any guidance as to why one difference approximation might be preferable to another. Instead, a lot of space is taken up by a detailed and repetitive discussion of how the replacement of differential operators by finite differences leads to systems of algebraic equations. Frequently, numerical values for the coefficients are given for some specific mesh-size. ...

This is odd in a book which claims to be a survey. For several decades, the more conventional marching procedures have been used, often with great success, by an enormous number of people. The basic algorithmic ideas behind these methods, as well as a simplified stability theory, could have been presented easily, even to an audience which is not very sophisticated mathematically.
...

The author is well known as a master of the very sophisticated art of obtaining numerical solutions to difficult applied problems. But this book does not fulfil the promise indicated in the preface, because it concentrates on his own work and neglects too many important methods.

4.2. Review by: Arthur O Garder Jr.
Mathematical Reviews MR0234642 (38 #2958).

This book is a very readable survey of numerical methods for solving partial differential equations. ... Certain partial differential equations, such as the non-linear equation for the soap film problem, are solved by means of a formulation in terms of the equivalent functional, using the calculus of variations. The functional is approximated by a discrete sum and the sum is optimised. The most difficult problem (from the numerical point of view) is the simultaneous solution of two mildly non-linear Navier-Stokes equations for two dimensional steady-state flow. An over-relaxation iterative technique is used to solve the difference equations. In all other cases the solution of the difference system is obtained by the generalised Newton method.

Problems which require more subtle numerical techniques are avoided here. Much of the current engineering literature is concerned with intricate non-linear systems of partial differential equations. Such problems require more emphasis on numerical techniques and their convergence properties ...

5.1. Review by: Anon.
The Military Engineer 60 (395) (1968), 242.

While designed as a first course in calculus, the author states that this is a mathematical book in which calculus is developed with the same rigour accorded to the other mathematical subjects. The organisation, presentation, and philosophy of the text are based on the assumption that the physical value of a mathematical model depends, in the last analysis on experimental verification; and, from the experimental point of view, nature is discrete.

6.1. Review by: Graeme Fairweather.
Mathematical Reviews MR0272147 (42 #7028).

As the title indicates, this is an introductory text in numerical analysis. In the first six chapters the usual topics, approximation, numerical integration and differentiation, systems of algebraic equations, and ordinary differential equations are considered. Chapters 7 and 8 are devoted to a brief study of two rather unusual topics, discrete model theory and interval analysis.

... in such a short book the author tries to consider too many topics and treats none in any detail. Only the simplest numerical methods are discussed, and the treatment is extremely elementary. The author does pay some attention to the theoretical aspects of the methods and on occasions proves existence and uniqueness theorems and derives error estimates.
...

For deeper studies of the topics mentioned in the book, the author provides the reader with an extensive list of references consisting of approximately 85 research papers and books. In the reviewer's opinion, most of the references are far too advanced to be of use to the average reader.

The choice of material leaves much to be desired. ...

The reviewer would hesitate to recommend this as a suitable text for any type of introductory numerical analysis course. The mathematically oriented student will find little of interest and may find statements like "From the computing point of view, a set of calculations that results in overflow is called unstable. Otherwise, a computation is called stable" objectionable. On the other hand, few of the methods discussed are of much practical use to the science or engineering student.

7.1. Review by: Stephen Castell.
The Mathematical Gazette 58 (404) (1974), 152-153.

This book is not, as the spoken title might indicate, a directory of well behaved, diffident, rather reserved displayers of fashionable clothes, but at times it is equally revealing. It is a book in the Courant-Fredrichs-von Karman-Taub-Truesdell-Carrier-Henrici-Belman-Kalaba tradition. That is, that brand of delightful and peculiarly American applied mathematics which rejoices in the analytic niceties of matrices, vectors, tensors and field theory but at the same time faces up to the real, even social, problems that applied mathematics is obliged to have a hand in and thereby achieves a meaningful contribution to the study of their numeric computer-implementation.

As such, the prose style tends to be in part a curious mixture of the philosophical, the quaintly classical and the idiomatic up-to-date.

But no matter. Donald Greenspan comes straight to the point in Chapter I: the majority of real problems in which applied mathematics has an interest are characterised by three things:

1. The realisation that the concept of infinity and related pure mathematics concepts like limit, integral, etc. are, practically, irrelevant.
   
   
2. Most of the problems are somewhere, somehow non-linear.
   
   
3. Sooner or later you are going to need a computer to solve them.

Moreover, since any validation of models developed to solve these problems is going to depend on measurement and thus discrete data, why not formulate discrete models in the first place? And so we are introduced to a rigorous theory of discrete models quite painlessly (and for those familiar with the computer concepts of addressability, storage locations, iteration and recursion, one might say easily and simply). Conservation principles are derived and there is a genuine attempt to make the true (rather than applied) mathematician feel it is worth his while reading on. On the other hand, by expressing many of the approaches in terms of the idea of an algorithm fairly early on, the power of the "if it works, program it" outlook is clearly revealed.
...

The book is one in the Applied mathematics and computation series and I hope that a follow-up is intended, showing how the models are significantly (and, therefore, computationally) better than others we are more used to and illustrating how economical in computer time they are with reference to some real industrial problems.

However, a potentially fruitful source of ideas for the industrial mathematician, and for the academic and the teacher there is much to delight and to be used for illustrating the contemporary relevance of the computer.

7.2. Review by: Alexander R Gourlay.
Mathematical Reviews MR0366132 (51 #2382).

The author proposes a discrete model theory as an approach to the understanding of diverse phenomena. Examples of its application include vibration, gravitation and fluid mechanics.

7.3. Review by: E V K.
Current Science 43 (5) (1974), 163.

This book is an interesting addition to the series of monographs on Applied Mathematics and computation. It is concerned with a new type of mathematical modelling where, a discrete approach is followed which is more amenable to computer analysis than the current physical theories.

The book consists of ten chapters and some interesting appendices.

Chapter I: Fundamentals of discrete model theory.
Chapter II: Discrete oscillators.
Chapter III: Nonlinear string vibrations.
Chapter IV: Planetary motion and discrete Newtonian gravitation.
Chapter V: The three-body problem.
Chapter VI: The $n$-body problem.
Chapter VII: Discrete fluid models.
Chapter VIII: Symmetry in discrete mechanics.
Chapter IX: Other forms of discrete mechanics.
Chapter X: Discrete special relativity.

Appendix A: The Van der Pol oscillator.

Appendix B: A discrete Hamilton's principle.

Appendix C: Conservation of Angular momentum in three dimensions.

An interesting set of research problems and good bibliography are provided at the end.

In conclusion, I feel that this book will be of interest to theoretical physicists, Computer Scientists and applied mathematicians. A one semester course on this subject will be of great value in Master's level Physics, Mathematics and Computer Science program.

8.1. From the Preface.

The development of the high-speed digital computer has had, and continues to have, a revolutionary effect on modern applied science. Immediate evidence is available in the form of a large number of computer-generated numerical solutions of fundamental, unsolved systems of mathematical equations. The diversity of fields being affected includes lunar and planetary astrodynamics, wave diffraction, shock waves, laminar flow of liquids, free-surface fluid flow, weather prediction, thermodynamics, elasticity, electrostatic and gravitational potential, optimal control, n-body problems, vibration theory, molecular interaction, quantum theory, and relativistic collapse. Less obviously, there have been natural, qualitative changes in related mathematical models and theories.

This book attempts to develop a broad spectrum of applications that can be formulated as problems in differential equations in the real domain. Existing analytical theories and techniques will be summarised appropriately so that the reader will understand when he should not use a computer. For those problems which cannot be solved analytically, we will develop finite difference, computer-oriented numerical methods for approximating solutions. Indeed, if a computer algorithm is defined as a finite sequence of computer operations designed to yield an approximate solution of a given mathematical problem, then this book is concerned primarily with the development of computer algorithms. In this connection, it must be understood that the immense power of the modern digital computer lies in its ability to perform arithmetic operations and to store and retrieve numbers with exceptional speed.

In order to develop in the reader the intuition which will enable him to devise sound, economical methods for his own particular problems, heuristic arguments are emphasised throughout. Sources for the precise mathematical foundations are referenced appropriately for the reader with a mathematically oriented background.

Finally, a few words are in order about the emphasis on difference techniques. It is at times possible, of course, to utilise a continuous method of approximation which, by some criterion, is superior to a finite difference method. Nevertheless, I have never seen an appropriate difference method fail where a continuous method works, and I have seen difference methods work where continuous methods have failed. The latter is especially noticeable in studies of the Navier-Stokes equations. This tremendous breadth of applicability and its inherent structural simplicity are what make difference methods so exceptionally valuable in any direct, numerical approach to problems of applied, scientific interest.

8.2. Review by: John M Perry.
SIAM Review 18 (1) (1976), 148-149.

This book is essentially concerned with finite-difference techniques for the solution of partial differential equations arising in problems of physics and engineering. It is in the style of a textbook, heavily dependent on illustrative examples, and includes 71 end-of-chapter exercises, largely routine; answers are provided for about one-third of them. The book is published by a photo-offset process, with a resulting awkwardness of display of equations, and thus is not as extensive a treatment as the 312 pages might imply. Typographical errors are fairly frequent but rarely intrude on the mathematical accuracy. The mathematical treatment of the techniques is highly heuristic, and there is infrequent mention of error bounds.

The level of mathematical background expected includes ordinary differential equations, some understanding of partial differential equations, and fairly routine matrix algebra. An acquaintance with computer techniques is also expected, there being frequent references to computational efficiency (usually in terms of the UNIVAC 1108) and many exercises expecting access to a computer. There is no specific discussion of programming, however, and a FORTRAN listing for a Navier-Stokes problem in an appendix is the only program shown.
...

Although not a distinguished book, it is very readable and quite suitable for its rather narrow purpose.

8.3. Review by: Gertrude Blanch.
Mathematical Reviews MR0362905 (50 #15343).

The introductory chapter deals with such matters as an elementary notion of matrices, the Gauss elimination technique, and the generalised Newton's method of iteration. There are examples and exercises in every chapter.

Chapter II is devoted to the approximate solution of ordinary differential equations, including both initial and boundary value problems. Chapters III, IV, and V deal, respectively, with elliptic, parabolic, and hyperbolic equations.

In Chapter VI, the classical variational problem in one and two dimensions is examined. Again the numerical solutions proposed depend on a difference-method approach. The author also deals briefly with variational problems associated with some partial differential equations, and with the Plateau problem in particular.

Chapter VII is devoted to some problems in fluid dynamics. Chapter VIII gives some notions of discrete model theory. There are two appendices and a Fortran program for a Navier-Stokes equation. Finally there is a list of references, answers to some exercises, and an index.

Considering the variety of topics dealt with in such a small book, it is not to be expected that any one topic will be treated in depth. However, the text well meets the author's aims for an elementary presentation of numerical methods suitable for a variety of problems in engineering and physics. The presentation is clear. Reproduction is by photo-offset of the typed manuscript.

9.1. From the Publisher.

Arithmetic Applied Mathematics deals with the deterministic theories of particle mechanics using a computer approach. Models of classical physical phenomena are formulated from both Newtonian and special relativistic mechanics with the aid only of arithmetic. The computational power of modern digital computers is highlighted, along with simple models of complex physical phenomena and solvable dynamical equations for both linear and nonlinear behaviour. This book is comprised of nine chapters and opens by describing an experiment with gravity, followed by a discussion on the two basic types of forces that are important in classical physical modelling: long range forces and short range forces. Gravitation and molecular attraction and repulsion are considered, along with the basic concepts of position, velocity, and acceleration. The reader is then introduced to the $N$-body problem; conservative and non-conservative models of complex physical phenomena; foundational concepts of special relativity; and arithmetic special relativistic mechanics in one space dimension and three space dimensions. The final chapter is devoted to Lorentz invariant computations, with emphasis on the arithmetic modelling and analysis of a harmonic oscillator.

This monograph will be of interest to mathematicians, physicists, and computer scientists.

9.2. From the Preface.

In this book we will develop a computer, rather than a continuum, approach to the deterministic theories of particle mechanics. Thus, we will formulate and study new models of classical physical phenomena from both Newtonian and special relativistic mechanics by use only of arithmetic. At those points where Newton, Leibniz, and Einstein found it necessary to apply the analytical power of the calculus, we shall, instead, apply the computational power of modern digital computers. Most interestingly, our definitions of energy and momentum will be identical to those of continuum mechanics, and we will establish the very same laws of conservation and symmetry. The unifying concept will be that of the potential. In addition, the simplicity of our approach will yield simple models of complex physical phenomena and solvable dynamical equations for both linear and nonlinear behaviour. The price we pay for such mathematical simplicity is that we must do our arithmetic at high speeds.

9.3. Contents.

Chapter 1. Gravity

1.1. Introduction
1.2. Gravity

Chapter 2. Long and Short Range Forces: Gravitation and Molecular Attraction and Repulsion

2.1. Introduction
2.2. Gravitation
2.3. Basic Planar Concepts
2.4. Discrete Gravitation and Planetary Motion
2.5. The Generalized Newton's Method
2.6. An Orbit Example
2.7. Gravity Revisited
2.8. Classical Molecular Forces
2.9. Remark

Chapter 3. The $N$-Body Problem

3.1. Introduction
3.2. The Three-Body Problem
3.3. Conservation of Energy
3.4. Solution of the Discrete Three-Body Problem
3.5. Center of Gravity
3.6. Conservation of Linear Momentum
3.7. Conservation of Angular Momentum
3.8. The $N$-Body Problem
3.9. Remark

Chapter 4. Conservative Models

4.1. Introduction
4.2. The Solid State Building Block
4.3. Flow of Heat in a Bar
4.4. Oscillation of an Elastic Bar
4.5. Laminar and Turbulent Fluid Flows

Chapter 5. Nonconservative Models

5.1. Introduction
5.2. Shock Waves
5.3. The Leap-Frog Formulas
5.4. The Stefan Problem
5.5. Evolution of Planetary Type Bodies
5.6. Free Surface Fluid Flow
5.7. Porous Flow

Chapter 6. Foundational Concepts of Special Relativity

6.1. Introduction
6.2. Basic Concepts
6.3. Events and a Special Lorentz Transformation
6.4. A General Lorentz Transformation

Chapter 7. Arithmetic Special Relativistic Mechanics in One Space Dimension

7.1. Introduction
7.2. Proper Time
7.3. Velocity and Acceleration
7.4. Rest Mass and Momentum
7.5. The Dynamical Difference Equation
7.6. Energy
7.7. The Momentum-Energy Vector
7.8. Remarks

Chapter 8. Arithmetic Special Relativistic Mechanics in Three Space Dimensions

8.1. Introduction
8.2. Velocity, Acceleration, and Proper Time
8.3. Minkowski Space
8.4. 4-Velocity and 4-Acceleration
8.5. Momentum and Energy
8.6. The Momentum-Energy 4-Vector
8.7. Dynamics

Chapter 9. Lorentz Invariant Computations

9.1. Introduction
9.2. Invariant Computations
9.3. An Arithmetic, Newtonian Harmonic Oscillator
9.4. An Arithmetic, Relativistic Harmonic Oscillator
9.5. Motion of an Electric Charge in a Magnetic Field

Appendix 1 Fortran Program for General $N$-Body Interaction

Appendix 2 Fortran Program for Planetary-Type Evolution

References and Sources for Further Reading

Index

9.4. Review by: Harry Hochstadt.
SIAM Review 24 (1) (1982), 100.

The author's avowed purpose is to "develop a computer, rather than a continuum, approach to the deterministic theories of particle mechanics. Thus, we will formulate and study new models of classical physical phenomena from both Newtonian and special relativistic mechanics by use of only arithmetic. At those points where Newton, Leibniz, and Einstein found it necessary to apply the analytical power of the calculus, we shall, instead, apply the computational power of the modern digital computers." It is also stated that this textbook is aimed primarily at undergraduates. In the brief space of some 160 pages the author discusses applications of his arithmetic approach to such problems as heat transfer, elastic vibration, laminar, turbulent and free surface fluid flow, planetary evolution, shock wave development, interface problems, and relativistic oscillation.

The gain of his method is, as he states, avoidance of calculus and differential equations. The discrete models developed can be programmed directly, and presumably these techniques can be taught to and used by people who have no knowledge of calculus. Is that a gain, however? It is doubtful that students who have not been exposed to calculus will be able to master the complexities of the notation and will be able to develop their physical and mathematical intuition from these models. There is no doubt that numerical answers to complex physical problems are hard to come by without recourse to high speed computers. But are we really ready to give up the elegant tools of calculus that were developed by such giants as Newton, Leibniz and Einstein? They not only please aesthetically, but they provide deep insight into the natural phenomena we study. Simple models that can be solved explicitly give us a deep understanding, which is then enriched by further numerical studies of more complicated models. The existence and uniqueness theorems of mathematical physics give us the confidence that what we are doing makes sense.

The author has performed an interesting tour de force in showing us how a more elementary arithmetic attack on these physical problems can be developed, which leads directly to numerical answers via a high speed computer. But, at best, such an approach should be developed in tandem with a study of the classical tools of mathematics and their application to the study of mathematical physics.

9.5. Review by: Robert B Kelman.
Mathematical Reviews MR0590428 (82d:70003).

The author has persistently pursued the concept that the laws of mechanics can be reformulated discretely, i.e., in the discrete/arithmetic of digital computers, rather than the arithmetic of real numbers. This obviously is not the first reformulation of mechanics, as one recalls Maupertuis, Lagrange, and Hamilton. The questions forced on each of these revisions are: Does it provide the ability to solve new problems? To solve old problems more easily? To obtain new insights? Is it easier to learn? The answer to the first two of these questions is obviously no, because continuum equations can be discretised and one of the principal objectives of numerical analysis is to show that there is an equivalence between the discretised equations and the continuum equations, i.e., as the norm of discretisation approaches zero the norm of the difference of the solutions of the two systems approaches zero.

There is no empirical evidence available from controlled pedagogical experiments as to whether or not students would learn classical mechanics more efficiently by the strictly discrete approach or by the classical approach. In fact, as most teachers now employ a "mixed" strategy, it may be increasingly difficult to obtain such data. The book does not contain problems, which limits its usefulness as the main textbook in a course. Its clear style and consistent viewpoint make it ideal supplementary reading. Topics include particle mechanics through the n-body problem, heat transfer including the Stefan problem, fluid dynamics, and special relativity.

9.6. Review by: J Astin.
The Mathematical Gazette 65 (434) (1981), 306-307.

This is an interesting though somewhat unusual book. Although it has a preface, it gives no indication of the level of sophistication needed by the reader, nor indeed does it indicate in any way to what group of students the book is aimed.

It is usual in numerical methods in applied mathematics to take a continuous model and then to approximate it by some type of difference scheme. The author in this book seeks to avoid this approach by building his initial model as a set of discrete states ...

This is a book to be read, digested, used and enjoyed.

10.1. From the Introduction.

Many mathematicians consider Albert Einstein to be a physicist, while many physicists consider him to be a mathematician. The reasons for such contradictory feelings lie in the same individual differences which prevent human beings from agreeing uniformly on almost anything, let alone on such complex, technical questions as to what is a mathematician or what is a physicist. nevertheless, it will be to our advantage to have some feeling, or intuition, about the nature of our subject, physics, before we begin our study formally. It is the aim of this chapter to develop such intuition and a convenient starting point is to try to understand the often used, rarely understood, statement that "mathematics is an exact science."

10.2. Review by: N Gass.
Mathematical Reviews MR0616638 (82h:70001).

This book is an introductory text to some topics in mathematical physics. It represents, however, a new approach since the models of physical phenomena discussed are described by using only arithmetic instead of calculus. Thus, the definitions of energy and momentum will be identical to those of an n-particle system (discrete models) in continuum mechanics but have the advantage of mathematical simplicity.

This book emphasises Newtonian mechanics including topics in gravitation, oscillations, waves, heat and fluid flows and planetary motions. Five FORTRAN codes selected from the above fields are appended to demonstrate the computational procedure.

11.1. From the Preface.

In this monograph, we develop and apply a family of powerful finite difference methods for the numerical solution of free boundary problems for incompressible fluids. The power is derived by combining advantageous aspects of the MAC method [T E Welch, F H Harlow, J P Shannon and B J Daly, "The MAC method: a computing technique for solving viscous, incompressible transient fluid flow problems involving free surfaces", LASL Rep. LA-3425, Los Alamos Sci. Lab., Los Alamos, N.M., 1966] for incompressible fluids, the Courant-Isaacson-Rees method for hyperbolic equations, successive overrelaxation iteration, and several special techniques developed by the authors. We show how problems of great complexity in both two and three dimensions can be solved quickly and economically on modern digital computers.

Our methods are called pressure methods because they solve for the pressure and the velocity variables together. Our particular algorithm solves for these simultaneously. Physically, this is important because separation of the pressure from the velocity, in the usual way, may result numerically in the transformation of an incompressible fluid into one which is compressible.

The book is structured in the following way. We begin by summarising those systems of fluid equations which are of broad general interest. Next, the basic numerical method is described, applied, and analysed for the simplest of these systems. Thereafter, as one progresses in the reading, one finds the systems studied in each chapter to be of greater complexity than those studied in the previous chapter. In the last chapter we provide the computer codes for all the systems studied.

12.1. Review by: Mohamed E El-Hawary.
https://dl.acm.org/doi/book/10.5555/43403

This numerical analysis textbook is suitable for a junior or senior undergraduate or first-year graduate introductory course. Familiarity with computer programming and ordinary differential equations is assumed. The authors' objective is to develop numerical analysis in such a fashion that the reader will be able to apply the techniques within a reasonably short time. The preface states that a major goal in writing the book was to develop a methodology in which the numerical solution for a given problem has the same qualitative behaviour as the analytical solution. The text begins in chapter 1 with the basics of algebraic and transcendental systems. The introduction describes the importance of numerical analysis. Next, matrices and linear systems, general linear algebraic systems of equations, Cramer's rule, diagonal dominance, and tridiagonal systems are defined.

Section 1-3 covers one practical method of solution: Gaussian elimination. In the next section, tridiagonal and diagonally dominant systems with negative diagonal and positive super- and sub-diagonal elements are shown to be nonsingular. Forward and backward solutions and LU decomposition are also discussed. The generalised Newton's method (starting with the classical version) is treated in Section 1-5. The over-relaxation factor is introduced and one- and n-dimensional problems are discussed. Mildly nonlinear systems are defined as linear systems in which each equation has a term nonlinear in only one variable. It is shown that if the linear part is tridiagonal and diagonally dominant, then the solution is unique. The authors make some remarks on the generalised Newton's method related to the choice of over-relaxation factor. In Section 1-7, entitled "Eigenvalues and Eigenvectors," the authors explore the power method as a numerical technique to find eigenvalues and eigenvectors. The end of the chapter includes 28 exercises, divided into 14 basic and 14 supplementary problems.

Approximation is treated in chapter 2. A one-paragraph introduction is followed by Section 2-2, where discrete functions are defined. Piecewise linear interpolation is introduced next. In Section 2-4, piecewise parabolic interpolation formulas are defined. Because piecewise linear and parabolic interpolation are not usually applicable to extrapolation, the cubic spline interpolation formula is delineated in Section 2-5. This section also contains a discussion of extensions to the concept. The general Lagrange interpolation formula of degree k is treated next, and least squares curve fitting rounds off the chapter. There are 16 basic exercises and 11 supplementary exercises in this chapter.

Approximate integration and differentiation are covered in chapter 3. The introduction discusses theoretical antiderivatives and leads to a definition of the trapezoidal rule; the corresponding error bounds are given. Simpson's rule, which is more accurate than and involves the same amount of computation as the trapezoidal rule, is discussed as the second integration technique. The higher-order method of Romberg integration is covered in Section 3-4. In the section entitled "Remarks About Numerical Integration," other related techniques are referenced. Numerical differentiation, which is required for differential equations, is introduced to conclude the chapter. The treatment includes error formulas. The chapter contains 13 basic and 12 supplementary exercises.

The approximate solution of initial value problems for ordinary differential equations is the subject of chapter 4, which begins with an introduction defining initial value problems. The simplest (Euler's) method dealing with one equation in one unknown is treated, and the convergence of Euler's method and roundoff errors are handled next. A second-order Runge-Kutta method to improve accuracy is discussed in Section 4-4. Extensions to higher-order Runge-Kutta formulas are covered next. Kutta's fourth-order method for a system of two first-order equations is treated in Section 4-6. Kutta's fourth-order formulas, and the method of Taylor expansion, for second-order differential equations are discussed in Sections 4-7 and 4-8. Instability manifested by overflow is then discussed, and potential remedial steps are described. In Section 4-10, dealing with the approximation of periodic solutions of differential equations, the authors use Van der Pol's equation as an example. The concluding section includes hints and remarks on future reading. The chapter is followed by 18 basic and 9 supplementary exercises.

The approximate solution of boundary value problems for ordinary differential equations is covered in chapter 5. In the introduction the authors define boundary value problems physically. Next, the central difference method for linear boundary value problems is introduced; a simple example that shows that a tridiagonal system arises is used to discuss issues of the existence and uniqueness of solutions to the problem. Because of the limitations of the central difference method, the upwind difference method for linear boundary value problems is introduced in Section 5-3. A discussion of the pros and cons of both methods is also given. The numerical solution of mildly nonlinear boundary value problems is treated in Section 5-4, along with conditions for the existence and uniqueness of solutions. As usual, examples help to clarify the concepts. Conditions of convergence of difference methods for mildly nonlinear boundary value problems are discussed in Section 5-5. The fundamental problem of the calculus of variations is discussed and Euler's differential equation is derived to introduce the finite element method in Section 5-6. Finally, differential eigenvalue problems are discussed briefly. The exercise section contains 15 basic and 8 supplementary problems.

Chapter 6 deals with elliptic equations. The introduction (Section 6-1) explains that a large portion of mathematical physics is devoted to the study of the elliptic class of partial differential equations. It discusses linear and mildly nonlinear problems and their classification as elliptic, parabolic, ord hyperbolic. Section 6-2 discusses boundary value problems for Laplace's equation, which is the simplest elliptic equation. A solution in terms of harmonic functions is included. Formulation of the Dirichlet problem concludes the section. The approximate solution of the Dirichlet problem on a rectangle and on a general domain is treated in Sections 6-3 and 6-4. The general linear elliptic equation is given on the basis of solution methods for the Dirichlet and central difference problems. Formulas are used to approximate the variables of interest in Section 6-5. An upwind difference method for general linear elliptic equations is then given. In Section 6-7, convergence theory for the numerical solution of linear boundary value problems is treated. The numerical solution of mildly nonlinear problems is examined in the concluding section of this chapter. There are 18 basic and 5 supplementary exercises at the end of the chapter.

Parabolic equations are the subject of chapter 7. In the introduction the authors indicate that the heat equation is a prototypical parabolic differential equation. A simple explicit numerical method for the heat equation is given in Section 7-2. The general linear parabolic equation is defined and a numerical method is formulated in Section 7-3; this is followed by an explicit upwind method. A discussion of the numerical solution of mildly nonlinear problems is offered next. The convergence of explicit finite difference methods is treated in Section 7-6. In the next section, the implicit central difference method is discussed for improvements such as high accuracy in space. Section 7-8 is devoted to an implicit upwind method. The chapter concludes with an exposition of the Crank-Nicolson method to improve time accuracy. Eighteen basic and four supplementary problems are given at the end of the chapter.

Hyperbolic equations are discussed in chapter 8. An introduction indicates particular difficulties and explains that the approach in this chapter, therefore, is to use either a second-order partial differential equation or the equivalent system of two first-order equations. The initial value or Cauchy problem and the initial-boundary problem are defined in Section 8-2. This is followed by an analytic solution to the Cauchy problem. Subsequently, an explicit method for initial boundary problems is given in order to develop intuition with regard to difference approximations for the wave equation. With this background available a simple, implicit method for initial boundary problems is given in Section 8-4; this method requires the solution of a tridiagonal system on each row of grid points and is stable. A discussion of the extension to mildly nonlinear problems follows. In Section 8-6, hyperbolic systems are discussed. A quasilinear system of two first-order partial differential equations in two unknowns is discussed here, and the linear case is considered. The hyperbolic condition in terms of matrix eigenvalues is developed, and the problem is cast in normal form. In Section 8-7, the method of characteristics for initial value problems is discussed. The method is very accurate if applicable, but does not apply if the eigenvalues are not constants. Hence, the method of Courant, Isaacson, and Rees to resolve this issue is introduced in Section 8-8. The chapter concludes with the Lax-Wendroff method to solve a hyperbolic system of equations that arises in gas dynamics. There are 14 basic and 5 supplementary exercises at the end of the chapter.

The Navier-Stokes equations are the subject of chapter 9. In the introduction the authors indicate that the nonlinear behaviour is such that existence and uniqueness theory is not available. The governing dynamical equations are detailed in Section 9-2, and the general form of finite difference equations to solve the initial-boundary problem for the Navier-Stokes equations numerically is developed in the next section. The consideration of efficient computer programs to solve the resulting large system of equations is facilitated by using the pressure approximation. A solution algorithm to determine the pressure field is given in Section 9-5. For illustrative purposes the widely studied cavity flow problem is examined in Section 9-6. The stability of the method is considered in Section 9-7, which concludes the chapter. There are five basic exercises and four supplementary ones at the end of this chapter.

Chapters 1-5 can be used for a one-semester course, and the whole book would be suitable for a full-year course. Naturally, computer implementation by the student is essential. The book includes answers to selected exercises and a bibliography. Topics such as number systems, the secant method, Weddle's rule, Richardson extrapolation, and the methods of Graeffe and Milne do not appear, but including nonlinear equations is a good innovation.

12.2. Review by: Charles Saltzer.
Mathematical Reviews MR0946685 (89g:65004).

This textbook is suitable for advanced undergraduate or beginning graduate courses. The first half of the book is a text for a semester course dealing with the solution of linear and nonlinear systems of equations, approximation of functions, numerical quadrature and the solution of initial value and boundary value problems for ordinary differential equations. The second half is a text for a semester course in the numerical solution of partial differential equations of the second order, culminating in a careful treatment of the numerical solution of the Navier-Stokes equation in two dimensions. Some background in the theory of partial differential equations is included for motivation. This book should be a very useful text for courses in numerical analysis. A valuable feature is the emphasis on nonlinear problems.

13.1. From the Publisher.

In this book the author has tried to apply a little imagination and thinking to modelling dynamical phenomena from a classical atomic and molecular point of view. Nonlinearity is emphasised, as are phenomena which are elusive from the continuum mechanics point of view. FORTRAN programs are provided in the Appendices.

Contents: Introduction: Quasimolecular Modelling: What It Is and What It Is Not.

Quantitative Modelling: Falling Water Drops, Colliding Microdrops of Water, Crack Development in a Stressed Copper Plate, Stress Wave Propagation in Slender Bars, Melting Points of Atomic Solids.

Qualitative Modelling: Biological Self Reorganisation, Cavity Flow, Turbulent and Nonturbulent Vortices, Vortex Street Modelling, Porous Flow, Q Modelling Combustion.

Conservative and Covariant Modelling: Conservative Q Modelling, Relativistic Motion.

Appendices

Readership: Physicists and applied mathematicians.

13.2. From the Introduction.

Science is the study of Nature. We study Nature not only because we are curious, but because we would like to control its very powerful forces. Understanding the ways in which Nature works might enable us to grow more food, to prevent normal cells from becoming cancerous, and to develop relatively inexpensive sources of energy. In cases where control may not. be possible, we would like to be able to predict what will happen. Thus, being able to predict when and where an earthquake will strike might save lives, even though, at present, we have no expectation of being able to prevent a quake itself.

The discovery of knowledge by scientific means is carried out in the following way. First, there are experimental scientists who, as meticulously as possible, reach conclusions from experiments and observations. Since experimental conditions can never be reproduced exactly, and since no one is perfect, not even a scientist. all experimental conclusions have some degree of error. Hopefully, the error will be small. Then there are the theoretical scientists, who create models from which conclusions are reached, often using mathematical methods. Experimental scientists are constantly checking these models by planning and carrying out new experiments. Theoreticians are constantly refining their models by incorporating new experimental results. The two groups work in a constant check-and-balance refinement process to create knowledge. And only after extensive experimental verification and widespread professional agreement is a scientific conclusion accepted as valid.

Our concern in this book is with a new area of theoretical modelling which is called quasimolecular modelling, or more succinctly, Q modelling, or, less precisely, particle modelling. Though specifics will follow in later sections, we observe now, for the purpose of providing an overview, that quasimolecular modelling is the study of the dynamical behaviour of solids and fluids in response to external forces, the solids and fluids being modelled as systems of molecules or molecular aggregates, which interact in a fashion entirely analogous to classical Newtonian molecular interaction. The dynamical equations of Q modelling are large systems of second order, nonlinear, ordinary differential equations.

Note that for linguistic simplicity, the term molecule will, be used throughout as a generic term which includes both atom and molecule.

The primary differences between quasimolecular modelling and molecular mechanics modelling (Alder and Wainwright (1960); Hoover (1984)) can be describe as follows. The field of statistical mechanics combines the rules of statistics with the laws of Newtonian mechanics to describe quantitative, large scale properties of continuous solids and fluids from the most probable behaviour of constituent molecules. Primary goals of statistical mechanics are the derivation of microscopic thermodynamic properties relating to such quantities as temperature, stress, internal energy, and heat flow, and the derivation of equations of state which relate pressure, energy, volume and temperature. Molecular mechanics modelling is a computer approach applied directly to a small molecular subset of a given substance with the objective of confirming or modifying large scale statistical mechanics properties or equations. The major results in molecular mechanics have been equilibrium, that is, steady state, results. Q modelling, on the other hand, is concerned, primarily, with non-steady state phenomena and with variations in dynamical response due to variation of system parameters. In addition, Q modelling applies both to sets of molecules and to sets of molecules which have been aggregated into larger units called quasimolecules. It is through quasimolecular systems that Q modelling can be made to simulate exorbitantly large systems of molecules.

For linguistic ease, we will often use the term particle rather than quasimolecule. However, it must be noted that this usage of the term particle is different from the usage of others. Buneman et al. (1980) and Hockney and Eastwood (1981) use the term particle to represent an ion in a plasma. Amsden (1966) and Harlow and Sanmann (1965) use the term to represent a fluid point of positive mass which moves in accordance with mass, energy and momentum conservation properties which are incorporated in a system of partial differential equations in two space dimensions.

In the present context, the term particle will always mean an aggregate of molecules.

14.1. From the Preface.

Contemporary science teaches us that:

(1) All things change with time.

(2) All material bodies consist of atoms and/or molecules.

This book is concerned with computer simulation of scientific and engineering phenomena in a fashion which is consistent with principles (1) and (2), above. Our approach demands the approximate solution of initial value problems for systems of ordinary differential equations. The computers used for the examples to be discussed are the Digital Alpha275 personal computer and the Cray YMP/8.

The presentation is divided into three parts. The first part is concerned with mathematical, physical and numerical considerations and serves as a basis for the remainder of the book. The second part is concerned with the development of intuition, which is accomplished through extensive qualitative simulations and analyses. The third part is concerned with quantitative simulation of basic scientific and engineering phenomena. The penultimate chapter extends the approach to Special Relativity, but in a fashion which does not require previous study of the subject.

In general, Chapters 3-16 are independent of each other, so that the reader can study an individual application of interest without studying the other chapters of the presentation.

The approach developed here is distinctly different from that of classical continuum mechanics. Simulation is founded on discrete concepts only and is entirely consistent with modern theories of dynamical behaviour.

Finally, I wish to thank the World Scientific Publishing Company, Singapore, for allowing me to use freely in this book related materials from my earlier book Quasimolecular Modeling (1991).

14.2. Review by: Benedict J Leimkuhler.
Mathematical Reviews MR1468731 (98m:70019).

Because of their conceptual simplicity, natural formulation, and ease of discretisation, Newtonian particle systems are probably the most widely used models in chemistry and physics. Not surprisingly, many articles appear each year discussing methods, algorithms and implementations for these problems, usually tailored to a particular application or variation on the Newtonian framework.

The current text attempts to give a comprehensive foundation for particle simulations, beginning with "mathematical, physical, and numerical considerations'', then proceeding to survey a variety of applications, including solitons, fluid flow, crack formation, melting points, and special relativity. While the computational issues regarding any one of these topics could easily fill a book, the author's terse style and streamlined presentation makes short work of each application. For each, a classical Newtonian N-body problem is introduced, the discretisation and implementation are briefly described, and some simulation results (often in the form of graphics snapshots) are then presented.

The author's cursory treatment of both mathematical foundations and application details will lose many potential readers. Numerical analysts and others interested in numerical methods for mechanical dynamics will find that the brief mathematical and numerical methodology described in the first two chapters is concerned strictly with conservation of energy and momentum (and covariance under translations and rotations) in numerical simulation. There is no reference to the symplectic structure, to constrained dynamics, or to any of the other significant developments in mechanics since Poincaré. Virtually the only numerical integrators mentioned are the leapfrog method and an implicit energy-momentum method. Stability and convergence issues for numerical methods are not discussed.

At the same time, a chemist or physicist looking for new simulation tools will likely find the simplified particle framework too constraining. Few hints are given about proper discretisation of modifications of the Newtonian N-body problem such as constant variable molecular dynamics, dipole or rigid body systems, systems with magnetic fields, or particle systems coupled to the Poisson equation (such as arise in plasma simulations), and there is little effort to tie in with current research on molecular dynamics and related topics.
...

Because of the straightforward style and presentation and the lack of imposing mathematical barriers, the book may be of most use for undergraduate students in chemistry and physics looking for topics for simulation-based term projects.

15.1. Note.

No information found.

16.1. From the Publisher.

The study and application of $N$-body problems has had an important role in the history of mathematics. In recent years, the availability of modern computer technology has added to their significance, since computers can now be used to model material bodies as atomic and molecular configurations, i.e. as $N$-body configurations.

This book can serve either as a handbook or as a text. Methodology, intuition, and applications are interwoven throughout. Nonlinearity and determinism are emphasised. The book can be used on any level provided that the reader has at least some ability with numerical methodology, computer programming, and basic physics. It will be of interest to mathematicians, engineers, computer scientists, physicists, chemists, and biologists.

Some unique features of the book include: (1) development of turbulent flow which is consistent with experimentation, unlike any continuum model; (2) applicability to rotating tops with nonuniform density; (3) conservative methodology which conserves the same energy and momentum as continuous systems.

16.2. Contents.

1. Nonlinear Oscillation
2. Relativistic Oscillation, Perihelion Motion
3. The Fundamental Problem of Electrostatics
4. Toda and Calogero Hamiltonians
5. Shock Waves
6. Cracks and Fractures
7. Laminar and Turbulent Flow
8. Contact Angles of Adhesion
9. Fluid Bubbles
10. Rotating Tops
11. Biological Self Reorganization, Bouncing Elastic Balls
12. Solitons
13. Minimal Surfaces
14. Discrete Conservation Laws

16.3. Review by: Benedict J Leimkuhler.
Mathematical Reviews MR2081103 (2005h:70002).

This book offers a survey of the use of $N$-body systems in the modelling of physical problems. The organisation, as a set of case studies, and the inclusion of many numerical experiments will make the book helpful for planning student projects, e.g. as part of a course in computational methods for engineers. On the other hand, the presentation of mathematical issues is very limited, numerical analysis is given short shrift, and modern ideas in geometric integration, such as symplectic integration, are entirely overlooked. Thus the book may not be as useful for applied mathematicians.

The presentation of numerical methods is a little confusing in places as a fully implicit scheme is first introduced with much discussion of integral conservation and covariance, whereas nearly all the experiments with larger models appear to have been performed using the explicit leapfrog (Verlet) method, only presented briefly in an appendix, whose properties are not even touched upon.

The introduction of models is a little eclectic as we go from some examples from mathematical physics to others in crack propagation, fluids and molecular dynamics. Only the nitty-gritty of simulation is described, such as force laws and initialisation. In general the modelling details and nuances presented are not sufficient for a student to be able to access the state of the art literature in these application fields.

17.1. From the Preface.

Turbulence is the most fundamental and, simultaneously, the most complex form of fluid flow. By necessity, because of the myriad phenomena exhibiting turbulent behaviour, this monograph focuses primarily, but not always, on a single type of problem, cavity flow. However, because an understanding of turbulence requires an understanding of laminar flow, both will be explored.

Groundwork is laid by careful delineation of the necessary physical, mathematical, and numerical requirements for the studies which follow, and includes discussions of N-body problems, classical molecular mechanics, dynamical equations, and the leap frog formulas for very large systems of second-order ordinary differential equations.

Molecular systems are then studied in both two and three dimensions, while particle systems, that is, systems which use lump massing of molecules, are studied in only two dimensions.

All calculations are limited to those which can be done on a personal scientific computer, in our case a Digital Alpha 533, so that the methods can be utilised readily by others. Our choice of the Alpha 533 is motivated out of the desire to maximise accuracy and minimise computer time. This computer has a 64 bit word built into the hardware. Three-dimensional calculations, which are restricted to Chapter 5 only, required several 'tricks' in order to enable their completion in a reasonable time, and these will be described in Chapter 5.

Though molecular simulations are of interest in themselves, they are also completely consistent with the current surge of interest in nano physics and with our belief that the mechanisms of turbulence are on the molecular level. Nevertheless, extension into the large is also of great interest, and it is for this purpose that we develop particle mechanics.

Though Section 2.4 is essential reading for all the computations described in Chapters 2–7, these chapters are, in general, relatively independent.

Finally, it should be observed that very often velocity fields for various figures, throughout, had to be rescaled for graphical clarity.

18.1. From the Publisher.

This work meets the need for an affordable textbook that helps in understanding numerical solutions of ODE. Carefully structured by an experienced textbook author, it provides a survey of ODE for various applications, both classical and modern, including such special applications as relativistic systems. The examples are carefully explained and compiled into an algorithm, each of which is presented independent of a specific programming language. Each chapter is rounded off with exercises.

18.2. From the Preface.

The study and application of ordinary differential equations has been a major part of the history of mathematics. In recent years, new applications in such areas as molecular mechanics and nanophysics have simply added to their significance.

This book is intended to be used as either a handbook or a text for a one- semester, introductory course in the numerical solution of ordinary differential equations. Theory, methodology, intuition, and applications are interwoven throughout. The choice of methods is guided by applied, rather than theoretical, interests. Throughout, nonlinearity and determinism are emphasised.

Chapter 1 develops Euler's method and fundamental convergence theory. Chapter 2 develops Runge-Kutta formulas through the highest order avail- able, that is, order 10. Chapter 3 develops Taylor expansion methodology of arbitrary orders. Chapter 4 develops conservative numerical methodology. Chapter 5 is concerned with very large systems of differential equations, such as those used in molecular mechanics. Chapter 6 studies practical aspects of instability. Chapters 7 and 8 are concerned with boundary value problems. Chapter 9 presents, in an entirely self-contained fashion, fundamentals of special relativistic dynamics, in which the differential equations and the related constraints are truly unique. Chapter 10 is a survey with references of the many other topics available in the literature.

Flexibility is incorporated by providing programs generically. Computer technology is in such a rapid state of growth that the use of a specific programming language can become outdated in a very short time. In addition, the individual who wishes to use a graphics routine is free to use whichever is most readily available to him or her.

Relatively difficult sections are marked with an asterisk and may be omitted without disturbing the book's continuity.

Finally, it should be noted that this book contains materials of interest in engineering and science which are not available elsewhere. For example, Chapter 5 develops numerical methodology which conserves exactly the same energy, linear momentum and angular momentum as does a conservative continuous system.

18.3. Review by: John C Butcher.
Mathematical Reviews MR2294499 (2007k:65001).

Numerical methods for differential equations are indispensable items in the bag of tricks of anyone working in theoretical physics, applied mathematics or engineering. This book provides an introduction to many of the numerical algorithms and techniques in common use, in the context of practical applications.

The numerical methods featured in the opening chapters are the Euler method, Runge-Kutta methods and the Taylor series method. Because of its relative simplicity, it is possible to provide a more theoretical framework for the Euler method than for Runge-Kutta methods. The latter are presented in a historical style with methods introduced in increasing order of accuracy and increasing order of time. For both Euler and Runge-Kutta methods, practical computations are provided to highlight the relation between actual performance and theoretical accuracy. The Taylor series method is presented mainly through example problems. Linear multistep methods are touched on only in a chapter on miscellaneous topics towards the end of the volume.

A chapter motivated by physical problems, especially those on a nano scale, introduces techniques for the solution of large second-order systems. This is followed by a discussion of the use of conservation and covariance in the construction of numerical schemes for physical systems. Three further chapters with a computational emphasis deal with stability questions, the solution of mildly nonlinear tridiagonal systems and boundary value problems. The work on particle methods is re-visited in a relativistic setting.

The special perspective this book offers is a study of particle methods in physical simulations, on a small length-scale, with particular attention to conservation principles.